Relative Hyperbolicity and Bounded Cohomology

Let Γ be a finitely generated group and Γ′ = {Γi | i ∈ I} be a family of its subgroups. We utilize the notion of tuple (Γ,Γ′, X,V ′) that makes the statements and arguments for the pair (Γ,Γ′) parallel to the non-relative case, and define the snake metric dς on the set of edges of a simplicial complex. The language of tuples and snake metrics seems to be convenient for dealing with relative hyperbolicity. For tuples, the properties of being finitely generated, finitely presented (cf. [28, 29]), of type Fn, of type F , and of having fine triangles are defined. Fine triangles are the ones that are “thin with respect to the snake metric”. Call a pair (Γ,Γ′) hyperbolic if there is a finitely generated tuple (Γ,Γ′, X,V ′) with fine triangles and with X fine. We give a definition of relative hyperbolicity of Γ with respect to Γ′ which slightly generalizes the definition of Bowditch, and show that this notion coincides with hyperbolicity of the pair (Γ,Γ′). We describe the snake resolution Stς(Γ,Γ′), or the relative standard projective resolution. It is used to define both relative cohomology and relative bounded cohomology. We generalize the argument in [22, 23] to show that if (Γ,Γ′) is hyperbolic then Hb(Γ,Γ ′;V )→ H2(Γ,Γ′;V ) is surjective for all bounded QΓ-modules V . The same holds for bounded RΓmodules, bounded CΓ-modules, and Banach modules. Moreover, this statement extends to several characterizations of hyperbolicity of the pair (Γ,Γ′). A classifying space (Y, Y ′) for a pair (Γ,Γ′) is naturally defined. We prove that each non-zero real (relative) cycle of dimension at least 2 for a hyperbolic pair (Y, Y ′) has positive simplicial (semi)norm.